The particle ltering and their applications

Oppenheim, Georges a Philippe, Anne b de Rigal, Jean c

aUniversité Orsay (Equipe de Probabilité et Statistiques. Laboratoire de Mathématiques. bat 425. Centre d'Orsay. 91405 Orsay Cedex. France [email protected]), et Université de Marne la Vallée bUniversité Nantes, Laboratoire de mathématiques Jean Leray UMR CNRS 6629, 2 rue de la Houssinière - BP 92208 - F-44322 Nantes Cedex 3 cL'Oréal Recherche. Chevilly Larue. France

Abstract

Particle ltering is a Monte - Carlo simulation method designed to approximate non linear lters that estimate and track the state of a dynamic system. We present the general principle of these algorithms and show the wide domain of applications using some examples.

Key words: Dynamic system, , Non-linear ltering, Particle lter, Sequential Bayesian ltering.

1 Introduction, Examples and Problems

1.1 The goal

In chemometrics, research relating to time varying systems is ongoing, as re- cent articles show Chen et al (2004); Chen et al (2007); Shen et al (2006). Whether they are autonomous or controlled, these systems account for dy- namic evolutions in domains as varied as , of oil or pharmaceuticals, , surveillance. The simplest situations are linear. These are well understood, both theoretically and practically. The key word for these is 'the Kalman Filter' (around 1960) where realistic situations are modelled by linear equations.

The lter consists in estimating the conditional distribution of the partially observed state of a stochastic process from a sample path. The Kalman Filter solves this exactly and quickly for linear Gaussian dynamics. Outside these

Preprint submitted to Elsevier March 16, 2007 cases, and in non-linear situations, other methods such as the particle meth- ods are available. These are based on the Monte-Carlo simulation which ap- proaches and estimates the conditional distribution (weak approximation). They propagate a particle system over time. These domains are currently lead- ing to publication of numerous articles where the theoretical and practical as- pects are raised. See, for example, books Cappé et al (2005); Del Moral (2004); Doucet et al (2001) or articles Doucet et al (2000); Le Gland and Oudjane (2004); Kong et al (1994); Del Moral et al (2006); Crasan and Doucet (2002). This list is not exhaustive, see http://www-sigproc.eng.cam.ac.uk/smc/ for complementary references.

1.2 Dynamic system

1.2.1 Linear dynamic system

This is the representation of a physical system that evolves over time k. Evolu- tion of its state (Xk) is interesting and we wish to estimate and predict it. The reference for these systems is the described by a state equation and a measurement equation: for any k ∈ N  X State equation Xk = FkXk−1 + k Y Measurement equation Yk = GkXk + k

the state noise X and the measurement noise( Y are independent, • (εk )k∈N εk )k∈N standard white noise (sequence of uncorrelated random variables with zero ). The noise ( Y is independent of the state . εk ) (Xk) • the distribution of X0, called initial or prior distribution, is uncorrelated with the processes X and Y . (εk )k∈N (εk )k∈N • unknown parameters may be present in Fk,Gk and in characteristics of white noises X and Y . (εk )k∈N (εk )k∈N The rst equation is that of a Markov process called state process, (Xk)k∈N which is not completely measured and that we would like to estimate. Infor- mation on Xk comes from measurements Yk. We should remember that a Markov process is a stochastic process for which prediction of the future based on the present-and-the-past does not require the knowledge of the past. In other terms, conditional distribution Xn+1 given past states (Xj)j≤n is a function of Xn alone:

P (Xn+1 ∈ A|X0,X1,X2,...,Xn) = P (Xn+1 ∈ A|Xn).

The second equation gives information on state Xk based on the observed process . For a given state, we suppose that the random variables (Yk)k (Yk)k∈N

2 are independent.

The theory is greatly simplied by these hypotheses on dependence structures.

1.2.2 Non-linear dynamic system

Formulation of a non-linear system, modelled on the linear model is as follows:  X = f(k, X , X , ϑX ) State equation k k−1 k (1) Y Y Measurement equation Yk = g(k, Xk, k , ϑ ) where ϑX and ϑY are unknown parameters. More generally the dynamic sys- tem can be described by two sequences of operators:

(1) The sequence of operators describes evolution of states over (Qk)k∈N Xk time. Therefore, the operator Qk is a Markov transition kernel

Qk(x, dy) = P (Xk+1 ∈ dy|Xk = x) (2) (2) a sequence of operators (called likelihood operators) that de- (ψk)k∈N scribes the conditionnal distribution of observations given the (Yk)k∈N state (Xk)k∈N ψk(x) = P (Yk|Xk = x) (3) These formulae may appear abstract but they cover numerous examples, both simple and complicated, that can all be studied in the same way. We give two classical statistical models that can be expressed in the form of a dynamic system.

• The model: Y = θX + ε where Y represents the observa- tions, θ the parameter (or state) to be estimated and X the explanatory variables, can be expressed as a linear dynamic model by taking  θk = θk−1 State equation

Yk = θkXk + k Measurement equation

• The linear dierence equation models as autoregressive processes. Let us take the example of an AR(3), which is a solution for the equation

Yk = a1Yk−1 + a2Yk−2 + εk It can be described by the following system

        0 1 0 Yk−2 Xk+1 =   Xk +   where Xk =   a2 a1 k Yk−1    Yk = 0 1 Xk+1.

3 1.2.3 An example of a highly non-linear function: Biomass and sheries, Campillo et al (2006)

This is an estimation of sh stocks and evaluation of the impacts of capture on the animal biomass Bt. Let It be an index of abundance measured each year t. We get a quite complex non-linear function

 Bt = F (Bt−1, θ) exp(σW Wt) State equation

It = a5Bt exp(σV Vt) Measurement equation where

• F is an AR(3) type operator

F (Bt−1, θ) = St−1Bt−1 − a2St−1St−2Bt−2 + a3(1 − a4St−1)

−1 and St = (Bt − Ct)Bt • the captured animal bio-mass sequences (Ct) during the year t and (It) are observed

• (Wt) and (Vt) are two independent sequences of Gaussian white noise. • the constants aj are estimated by the algorithm. These equations show the type of problem that can be raised by particle methods. Here the diculty comes from the fact that the estimation is based on few noisy measurements taken over a long period of time.

2 How is it done?

2.1 The Kalman lter

The Kalman lter is the optimal lter for Gaussian linear dynamical mod- els (see Anderson and Moore (1979) for example). Because of its Markovian character, we can compute the conditional distributions given the state by recurrence:

1. Forecast step

p (xt/y0, . . . , yt−1) ← p (xt−1/y0, . . . , yt−1)

2. Filtering step

p (xt/y0, . . . , yt) ← p (xt/y0, . . . , yt−1) and yt

4 In a Gaussian context, these formulae are reduced to conditional expectations and

E (xt/yu, yu−1, . . . , y1) and Var (xt/yu, yu−1, . . . , y1) by taking u = t − 1 for the forecast step, and u = t for the ltering step.

There are numerous extensions to Kalman ltering, for example the extended Kalman lter (EKF) for non-linear models. The method is based on techniques of linearisation of the function around the current estimate. This lter is ef- fective if a reference trajectory is available and if the model is almost linear in the proximity of the reference trajectory or of the target. Theoretical proofs of stability are recent.

Outside the Gaussian domain, the problem can no longer be reduced to the computation of the two rst moments; the conditional distributions P (Xk/Yk = yk, ..., Y0 = y0) must be calculated or estimated. These probability distribu- tions describe what is interesting about the unknown state Xk, when the ob- servations yk, ..., y0 are known. The question is then: how can the conditional distribution P (Xk/Yk = yk, ..., Y0 = y0) be calculated and how does it evolve with time k as the observations are obtained?

2.2 Monte Carlo Method

Monte Carlo methods are simulation techniques for approximating distribu- tions or functionals of the distribution for example expectation and . Two large complementary families of algorithms are distinguished:

• Markov Chains for Monte Carlo (MCMC) techniques, • Monte-Carlo ltering techniques called particle ltering.

2.2.1 MCMC

MCMC algorithms are iterative methods for simulating a sample whose dis- tribution is the target π (possibly known up to a constant). The central tool is simulating the Markov chains with stationary distribution π. The simulated trajectories are then used to estimate integrals of the form ∫ h(x)π(x)dx (see Robert and Casella (2004)).

These methods are widely used in Bayesian to approximate the posterior distributions when the models' parameters are assumed to be con- stant over time (see Robert (2001)). The MCMC approximation can also be considered for problems of estimations linked to dynamic systems. For these

5 models, MCMC algorithms give an estimation of the conditional distribution

P (X1,...,Xk/Yk = yk, ..., Y0 = y0) and marginal distribution P (Xt,/Yk = yk, ..., Y0 = y0) for all t = 1, . . . , k. This is the classic situation of smoothing. These MCMC algorithms are not suited to dynamic models because the dis- tribution must be entirely recalculated each time that a new observation is obtained. In this situation, particle ltering is generally a good alternative to MCMC.

2.3 Particle Methods also called Particle Bayesian ltering

They group together sequential methods estimating a sequence of , in particular the conditional distribution P (Xk/Yk = yk, ..., Y0 = y0) for each instant k.

Particle ltering methods proceed like Kalman ltering by recurrence over time, by linking the two forecasting and ltering steps for each time step. Theses two steps no longer consist in calculating parameters of a Gaussian distribution (expectation and variance), but to a discrete approximation of the distribution of the state, requiring importance and .

This approach enables treatment of non-linear dynamic systems in particular. Furthermore, the noise can be non-Gaussian or not belong to an .

As for all the Monte Carlo methods, a sample is generated (1) (n) that xk , . . . , xk follows a distribution that approximates the conditional distribution P (Xk/Yk = yk, ..., Y0 = y0) then we estimate:

n the distribution by the empirical distribution 1 X where denotes • δ{x(i)} δ n i=1 k the Dirac measure. In other words, this is the uniform distribution on values generated. If the density of the distribution exists, it can be estimated by a kernel method. R • functionals of the form h(xk)p(xk|yk, . . . , y0)dxk by

1 X (i) nh(xk ). n i=1

The quadratic convergence rate of these estimators does not depend on the state space dimension.

6 2.4 The Particle lter algorithmic scheme

Before presenting the algorithmic scheme (the central point of this text), two remarks are necessary to explain the description of the algorithm which is summarised by gure 2.4.

Remark 1 The conditional distribution Pk = P (Xk/Yk = yk, ..., Y0 = y0) is unknown. We only know that it can be calculated iteratively. The structure of the equations of the model in its two aspects Markov/Likelihood described by equations (2) and (3) leads to the following recursive formula for the con- ditionnal distribution of X1,...,Xk given Yk, ..., Y0

p(yk/xk)p(xk/xk−1) pk(x1, . . . , xk/yk, ..., y0) = pk−1(x1, . . . , xk−1)/yk−1, ..., y0) p(yn/y0, ..., yn−1)

Forecast Correction Pk−1 −−−−→ Pk/k−1 −−−−−−→ Pk. Qk ψk where Pk/k−1 is the conditionnal distribution of Xk given Yk−1, ..., Y0). With this recurrence and if a of initialising the recurrence is available, at time k = 0, calculating Pk is feasible. Remark 2 Unfortunately, to do the previous calculations by hand happens to be extremely dicult. Several solutions exist, which vary from approximat- ing the ltering equations (see Durbin and Koopman (2001) for example) to particle simulation techniques (See Cappé et al (2005), Doucet et al (2001)). We estimate the distribution by ˆ(N) using particles. The general prin- Pk P k N ciple is as follows: we simulate random independent variables called particles whose weights evolve over time as a function of the values of the available observations.

The update is calculated when the value yk is observed. An iteration step is given by the following sequence in which (i) (i = 0, ..., N) represents the particle index. We assume here that the distributions have a density with respect to Lebesgue measure. The passage from ˆ(N) to ˆ(N) is as follows. P k−1 P k

Step 1 Simulation of particles according to the distribution ˆ(N) , the state N P k−1 of the particle is denoted ˜ (i) (i) Xk−1 Step 2 Transport of particles ( towards time of state ˜ (i) by i = 1, . . . , n) k+1 Xk−1 the transition associated with the state equation X Xk = f(k, Xk−1, εk ) producing a state ˜ (i). We are now at the instant Xk k. Step 3 Construction of a pseudo-observation ˜ (i) deduced from ˜ (i) and from Y k Xk the observation equation Y . Yk = g(k, Xk, εk ) The particles are now carrying the information pairs ˜ (i) ˜ (i) for (Xk , Y k )

7 ( N ) ä pk −1 = P( X k −1 /(Y1 = y1 ,...Yk −1 = yk −1 )

( N ) ( N ) Distribution pk −1 Distribution pk

Simulation Estimation

(i ) particles X À (i ) ŒÀ X = f (k , X ,ε ), Œ áX k á à k k −1 k X 0 - π 0 (i ) ( N ) Y particles à … á ÕŒYk = h(k , X k ,ε k ), (i ) … X k −1 - p k −1 á ÕŒY k

(i ) w k á (i) i = particle,i =1..N Y k k = time, Yk observation 1

Figure 1. The passage from ˆ(N) to ˆ(N). P k−1 P k i = 1, . . . , n. Step 4 Now the information from the value of observation yk at instant k is used. Comparison of each of the pseudo-measurements (i) with mea- y˜k surement yk is used to calculate the value of a comparison index, coef- cient (i) (i) PN (i) This weight, associated with each wk , wk ≥ 0, i=1 wk = 1. particle, is obtained by a kernel technique similar to that used for den- sity estimation (a kernel is a positive, often symmetrical function that has an integral of 1 for scalar variables and is often the tensor product of such functions for vector variables). Step 5 Density ˆ(N) remains to be estimated by basing ourselves on ˜ P k Xk(i) and the weight (i) by a kernel technique. wk

N ˆ(N) X (i) ˜ (i) Pk (x) = wk Kh(x − Xk ) i=1

−d where Kh(x) = h K(x/h) with K being a kernel, h the bandwidth of the kernel and d the dimension of the state space, the space to which Xk belongs. Step 6 The step is nished

Initialisation of this algorithm is done based upon a prior distribution Π0 which is used to simulate N particles at instant k = 0.

Numerous theoretical and practical information questions on this procedure are raised in the N.Oudjane's PhD (see Oudjane (2002); Le Gland and Oud- jane (2004)). Details on implementation of the algorithm, in particular usual problems linked to the choice of kernel K and to the convergence rate of

8 bandwidth h, are given in Rossi (2004).

Simulation according to the estimated distribution ˆ(N) is performed as fol- Pk lows:

1. We begin by simulating z according to the multinomial distribution of parameters (1) (N) wk , . . . , wk 2. Then we simulate ε according to the distribution admitting the kernel K as density. 3. We take z + hε

• What is the simulation procedure stopping rule?

This question is not crucial because the iterative procedure stops when no observation yk is available.

• How to choose N?

The choice of the number of particles N is also a tricky problem which is based on the estimated lter convergence properties. Oudjane (2002); Le Gland and Oudjane (2004) prove convergence of the estimated lter ˆ(N) towards the Pk unknown lter Pk, under reasonable conditions. The same result species the convergence rate: intuitively, if the Markov process mixes well and if the in- formation supplied by measurement on the state is good, N is small, on the order of a few thousand. These notions can be made technical by using norms for operators Ψ and Q.

• What happens if time k → ∞?

When certain state variables (denoted ϑ, here) do not vary over time, for example the parameters ϑX and ϑY in (1) or characteristics of white noises, we ˆ have results for convergence of sequence ϑt towards the true value of parameter ϑ∗. This follows from classical results on convergence of Bayes estimators (see Schwartz (1965)) which ensure the almost sure convergence of the posteriori ∗ expectation E(θk|y1, . . . , yk) towards θ and concentration of the conditional ˆ ∗ distribution of θk given y1, . . . , yk around θ .

• Is the algorithm well designed or does it degenerate?

If we estimate lter by empirical measure P (i) , the algorithm de- Pk wk δX˜ (i) generates in the sense that a great number of particlesk quickly reach zero weight. Dierent strategies have been proposed to evaluate algorithm de- generation (see Kong et al (1994)) and to correct this phenomenon. Numer- ous improvements have been proposed. They are essentially based on resam- pling/bootstrap techniques (see Doucet et al (2004); Gordon et al (1993) for instance) or on regularisation of the estimator. It is this second approach that

9 is adopted in the algorithm described in this paper.

• What information must we have to work with the algorithm?

The most important thing is to be able to simulate a value for state-observation pair (Xk,Yk) of the process of the instant k if the state xk−1 is known. This is the case, for example, if we have explicit functions f and g and the distrib- utions of noises X and X . One particularly simple and common case is the εk εk non-linear additive noise case X Y . Xk = F (Xk−1) + εk ,Yk = G(Xk−1) + εk

• What is the role of the initial Π0 distribution necessary to start the algorithm?

If the time with respect to internal time constants for evolution of the dynamic is long, then the eect of the initial distribution disappears quickly.

3 Applications

3.1 Tracking moving bio-cell, Shen et al (2006)

In biology research manual marking becomes dicult to carry out and very expensive when seeking to mark numerous moving cell and nucleus boundaries. Classic Kalman ltering methods need simplied models. They are not suited to sequences of images that include cell superposition (Isard et al (1998)). A specic lter enables automatic tracking. It is based on `Sequential Importance Sampling', one of the now classic simulation procedures (see Cappé et al, 2005, Chap. 7) . The chain of sets is robust, produces results of the same quality as a biologist but performs the work more quickly and can work for longer.

3.2 Polymerisation reactor, Chen et al (2003)

"A polymerization process in a continuous stirred tank reactor is a process in which monomer styrene, initiator and solvent are continuously fed into the reactor with constant ow rate and compositions. The system has eight state variables including concentration of initiator, solvent and monomer in the tank, temperature in the tank and the cooling device, and the three rst moments of the molecular weight distribution. The governing process model is non-linear." The dynamic system is based on discrete model. The equations and the operating conditions are given in Chen (2004) (see page 82-86).

10 3.3 Applications: can the initial distribution help in estimating?

We are interested in simplied electricity consumption, centred on the eect of temperature. These models are written as

+ Yt = αt(ut − Tt) + εt (4) where Tt represents temperature. It is an observed variable. The parameters are

• a heating gradient that is not observed and varies over time (αt) • a heating threshold (ut) (It represents the temperature above which heating is zero.) This parameter intervenes non-linearly in the regression model 2 • a variance σ of noise εt The equations of the state variables are written as

α and (α)2 (α)2 αt+1 = αt + t σt+1 = σt U and (U)2 (U)2 ut+1 = ut + t σt+1 = σt 2 2 σt+1 = σt where α and U are independent sequences of i.i.d. Gaussian variables with εt εt zero mean and variance σ(α)2 and σ(U)2 respectively. We wish to prove the inuence of the prior distribution (law at instant t = 0) on estimation of the distribution of interest. We simulate the observations according to the model (4) with constant parameters u = 15, α = 4 and σ = 4. The results obtained by particle ltering are summarised in Figure 2. We see that the gradient and threshold estimators are of suitable quality as far as the trajectories for the estimators vary around the nominal model. Beginning with very unfavourable prior distribution, the length of the transition period is around 40 days, similar for two estimators.

The estimator 2 underestimates the nominal value but the bias is (σt ) σ = 4 corrected over time, when the number of available observations increases.

The convergence of sequences (α)2 and (U)2 towards zero is classic when σk σk come from a constant parameter model.

3.4 Depollution of waste water according to J.P Vila and V.Rossi

This study, as the authors present it see Rossi (2004), relates to "ltering of a bioprocess by anaerobic digestion in a continuous waste water treatment bio- reactor". The interesting state variables, for example bacteria concentration,

11 Figure 2. Output of the particle ltering : evolution of the mean of conditional distribution ˆ(N) and its error as a function of the number of observations for the Pk dierent parameters. The dataset is a sample of observations simulated according to the model (4) with constant parameters u = 15, α = 4 and σ = 4. The results are derived from 50 independent samples are not measured and must be estimated. Filtering is therefore useful. Since the best model is not linear and cannot be made linear around a reference trajectory, particle ltering is required. The model took one year's work to construct. It is a little too complex to be described completely. It is constituted by a family of non-linear dierential equations, in which essentially rst order derivatives are involved. One of the equations species the law of bacterial growth.

Eight parameters (unknown values) add to the complexity. The variables are bacteria (B1, B2), strong ions, chemical oxygen need (S1), volatile fatty acid

(S2), and inorganic carbon concentrations. The rate of dilution and the CO2 ow complete the list, together with the command variable and substrate dilution.

Measurements are taken every 2 minutes for 30 days (21600 measurements). The prior distributions on parameters are selected independently and uni- formly over suitable intervals.

Exceptionally, a specialised and expensive sensor was included in the study, measuring values of the two states S1 and S2 over time. In operation , this measurement is not made. Several types of calculations are performed, with estimation made by experts or with on-line estimation. The results are good and improve by sequential estimation. The number of particles is 1000. The calculation duration is quite acceptable and is compatible with the phe- nomenon's evolution time constants.

12 References

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13 A. Kong, J.S. Liu, and W.H. Wong, Sequential imputations and Bayesian problems. JASA, Vol. 89, No. 425,278-288, (1994). F. Le Gland, N. Oudjane, Stability and uniform approximation of nonlinear lters using the Hilbert metric and application to particle lters. Ann. Appl. Probab. 14, No.1, 144-187 (2004). N. Oudjane, Stabilité et approximations particulaires en ltrage non linéaire - Applications au pistage PhD thesis, Université de Rennes 1, (2002). C.P Robert and G. Casella, Monte Carlo Statistical Methods, Springer-Verlag, New York. (2004). C.P. Robert, The Bayesian Choice: from Decision-Theoretic Motivations to Computational Implementation Springer-Verlag, New York. (2001). V. Rossi, Nonlinear ltering with convolution kernels. Application to a process of biological depollution. Ecole National Supérieure Agronomique de Mont- pellier 279 p. (2004). http://vrossi.free.fr/these.html V. Rossi, J-P. Vila, Nonlinear ltering in discrete time : A particle convolution approach. Ann. I.S.U.P., vol.50, No.3, p71-102 (2006). L. Schwartz, On Bayes Procedures, Z. Wahrscheinlichkeitstheorie 4, 10-26 (1965). H. Shen, G. Nelson, S. Kennedy, D. Nelson, J. Johnson, D. Spiller D, White M.R.H., Kell D.B. Automatic tracking of biological cells and compartments using particle lter and active contours. Chemometrics and Intelligent Lab- oratory Systems (2006).

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